Churchs Thesis and Humes Problem Justification as Performance
Church’s Thesis and Hume’s Problem: Justification as Performance Kevin T. Kelly Department of Philosophy Carnegie Mellon University kk 3 n@andrew. cmu. edu
Further Reading (With C. Glymour) “Why You'll Never Know Whether Roger Penrose is a Computer”, Behavioral and Brain Sciences, 13: 1990. • The Logic of Reliable Inquiry, Oxford: Oxford University Press, 1996. • (with O. Schulte) “Church's Thesis and Hume's Problem, ” in Logic and Scientific Methods, M. L. Dalla Chiara, et al. , eds. Dordrecht: Kluwer, 1997. • (with O. Schulte and C. Juhl) “Learning Theory and the Philosophy of Science”, Philosophy of Science 64: 1997. “The Logic of Success”, British Journal for the Philosophy of Science, 51: 2000. “Uncomputability: The Problem of Induction Internalized, ” Theoretical Computer Science 317: 2004. Computability and Learnibility, Textbook manuscript, 2005.
Fateful First Cut Relations of Ideas Matters of fact Analytic Synthetic Certain Uncertain A Priori A Posteriori Philosophy of Math Philosophy of Science Proofs and algorithms Confirmation Truth-finding “Rationality” Computability Probability
Painful Progress Relations of Ideas Matters of fact Analytic Synthetic Certain Uncertain A Priori A Posteriori Philosophy of Math Philosophy of Science Proofs and algorithms Confirmation Truth-finding “Rationality” Computability Probability
Slight Rearrangement Philosophy of Math Philosophy of Science Certain Uncertain A Priori A Posteriori Proofs and algorithms Confirmation Truth-finding “Rationality” Computability Probability
Philosophy of Science 101 Formal Questions Scientific Questions Unverifiable “The sun will never stop rising” Verifiable “The given number is even”
Wrong Twice! Formal Questions Unverifiable Scientific Questions Unverifiable Verifiable “The sun will never stop rising” Verifiable “The given number is even” “The next emerald is green” “The given computation will never halt”
Classical Excuse Relations of Ideas Completed analysis Rational ? Reason Matters of fact Unbounded experience Animal ? ? Always black? Observation
But Maybe… Relations of Ideas Unbounded experience Unbounded analysis ? ? Matters of fact Rational ? Reason ? ? Always black? Never white? Observation
Kant Synthetic A Priori 3 3 2 4 A Posteriori Unbounded experience 5 temporal intuition 3+2 = 5 ? Intuition ? ? Always black? Never white? Observation
Oops! Synthetic A Priori spatial intuition ? ? Never cross? Intuition A Posteriori Unbounded experience ? ? Always black? Never white? Observation
After Turing Formal Problems Unbounded computation Input Empirical Problems Unbounded reality n ? ? Never halts? Computer ? ? Always black? Never white? Observation
Bad First Cut Formal Questions Unverifiable Scientific Questions Unverifiable Verifiable “The sun will never stop rising” Verifiable “The given number is even” “The next emerald is green” “The given computation will never halt”
Good First Cut Formal Questions Unverifiable Scientific Questions Unverifiable Verifiable “The sun will never stop rising” Verifiable “The given number is even” “The next emerald is green” “The given computation will never halt”
Computational Epistemology For formal and empirical questions: Justification = truth-finding performance n. Find the truth in the best possible sense n. And then as efficiently as possible.
Verification as “Support” 1. 0 Deductive verification . 5 Inductive verification
Paradigms Evolve • Objective support by evidence.
Paradigms Evolve… • Objective support by evidence. • Coherent personal opinion. • “Rational” update.
Paradigms Evolve… • Objective support by evidence. • Coherent personal opinion. • “Rational” update.
But Leading Metaphors Matter • Myopic • Bedrock = “rationality” intuitions • Success = being “rational” • Truth-finding performance deferred or ignored • Problem complexity ignored • Little resemblance to computability
Verification as Halting Conclusion Data
Verification as Halting Conclusion Data Yes!
Verification as Halting Conclusion Data
Verification as Halting Conclusion Data
Verification as Halting ! g n Conclusion Data Ba
Verification as Halting Conclusion Data Can’t take it back.
Verifiability Yes! Conclusion true Conclusion false
Laws are Not Verifiable Always green?
Laws are Not Verifiable Always green?
Laws are Not Verifiable Always green?
Laws are Not Verifiable Always green?
Laws are Not Verifiable Always green? You must halt with “yes” if it’s true!
Laws are Not Verifiable Always green? You must halt with “yes” if it’s true!
Laws are Not Verifiable Yes! Always green?
Laws are Not Verifiable Always green?
Laws are Not Verifiable Always green?
A Purely Formal Problem Never shoots?
Analogy? I ain’t a’ shootin’, yeller belly! Never shoots?
Apparently Not Game’s up, J. R. ! Yer program’s showin’! Never shoots? 666
Apparently Not Now shaddup! I’m calculatin’ a-preeorey-like what yer gonna do. Never shoots? 666
Apparently Not ? !!! Never shoots? 666 23455 21456
But Then Again… I got it! Two kin play that opry oary game! Never shoots? 666 23455 21456
But Then Again… I won’t shoot ‘til I see him shoot in this here opry oary sim-yooo-laytion! Never shoots? 666 23455 21456
But Then Again… 666 23455 21456 I ain’t a’ shootin’. Never shoots? 666 23455 21456
But Then Again… 666 23455 21456 I still ain’t a’ shootin’. Never shoots? 666 23455 21456
But Then Again… 666 You check that there opry oary all you want… Never shoots? 23455 21456 666 23455 21456
But Then Again… 666 23455 21456 An’ all you’ll see is I ain’t a-shootin’ Never shoots? 666 23455 21456
But Then Again… Aw, shucks, J. R. I reckon You ain’t never gonna shoot. 666 23455 21456 Aw, shucks, J. R. I reckon you ain’t never gonna shoot. Never shoots? 666 23455 21456
But Then Again… 666 Ba ng ! 23455 Never shoots? 666 23455
But Then Again… 23455 Never shoots? 23455 666
But Then Again… 23455 666 See ya in the grand ol’ opry oary, sucker! 23455 Never shoots? 666
But Then Again… Ba Never shoots? 23455 ng ! Take that! 666
23455 Never shoots? 666 But Then Again…
But Then Again… ROBO SHERIFF 23455 Never shoots? 666 Fooled A priori
Rice-Shapiro Theorem Whatever a computable cognitive scientist could verify about an arbitrary computer’s input-output behavior by formally analyzing its program… 45863 Q D Qoc
Rice-Shapiro Theorem …could also have been verified by a behaviorist computer who empirically studies only the arbitrary computer’s inputoutput behavior! 45863 Q D Qoc
Uncomputable Induction H N Yes! Will see a non-halting machine Verifiable by “ideal agent”.
Uncomputable Induction H N ? Will see a non-halting machine Q Not verifiable by computable agent.
Similarity Halting Problem Universal Law Unverifiable Demon can fool computable agent Demon can fool ideal agent Answer runs beyond empirical experience Answer runs beyond formal experience
A Difference Halting Problem Universal Law Input ends Input never ends Can handle some examples a priori… Problem of induction “Internal” problem of induction eventually catches up!
Give and Take Formal input Empirical input Jaw Breaker Noodle Fits in mouth but may never melt May never fit in mouth May never swallow . . .
Another Difference? Formal Science Empirical Science Incompleteness Problem of induction
Not Necessarily Formal Science Empirical Science Incompleteness Problem of induction Add more powerful axiom Who knows if new axiom is consistent with old? Assume it is, keep checking, and hope Conjecture a theory Who knows if theory is consistent with data? Assume it is, keep checking, and hope
Thesis The problem of induction and the problem of uncomputability are essentially similar. So the two should be understood similarly. From the ground upward.
Philosophy: A House Divided • Unverifiability • “Confirmation” • “Support” • “Coherence” • “Rational” update, etc. • Uncomputability. • Suck it up • Logic vs. engineering • Try to approximate, etc. ? !
Unified Approach: Gun Control • Find the truth in the best feasible sense. • So drop the halting requirement if infeasible. Q
Start with Problems, Not Methods Partition Q over set K of inputs A Finite input B C D Infinite input
Convergence ? with halting Convergence ? in limit (in limit) ? ? A ? ? No more revisions ? A ? ? B C ? C C revisions . . .
Success Converge to right answer A B C D Don’t converge to wrong answer
Special Cases Verify A A Refute A B A Decide A A B B
Special Cases Compute partial f f f=0 f=1 f=2 . . . Dom(f) Theory selection M 0 M 1 M 2 . . .
Solutions and Optimality • Solution: method that succeeds on each input. • Optimal solution: solves in best possible sense • Solvable problem: has solution. • Problem complexity: best sense of solvability
Halting Bounds Revisions A B Verifiability convergence with 1 revision ending with “no”. Refutability convergence with 1 revision ending with “yes” Decidability convergence with 0 revisions.
Generalization to n Revisions A n-Refutation = n+1 rev ending with -A B n-Verification = n+1 rev ending with A n-Decision = n rev
Empirical Complexity Lim-ver Lim-ref Lim-dec. . . Underdetermination . . . and 1 -ver 1 -ref or 0 -ref closed 1 -dec 0 -ver open 0 -dec clopen
Formal Complexity. . . Lim-ver Lim-ref Lim-dec. . . Uncomputability and 1 -ver 1 -ref or 0 -ref Co-r. e. 1 -dec r. e. 0 -ver 0 -dec decidable
Formal + Empirical Complexity Lim-ver Lim-ref Lim-dec. . . Uncomputability + Underdetermination . . . and 1 -ver 1 -ref or 0 -ref Co-r. e. 1 -dec r. e. 0 -ver 0 -dec decidable
Ship-shape Epistemology. . . Effective empirical . . Ideal empirical Effective formal
Ship-shape Epistemology. . . Topological Recursive invariants
Ship-shape Epistemology. . . Lower Bound Combination Lower Bound
Messy, Odd Distinction. . . Mixed effective Borel . . Empirical Borel Formal arithmetical
Neat, Natural Distinctions. . . Mixed effective Borel . . Empirical Borel Formal arithmetical
Example: Kantian Antinomy Infinitely Finitely divisible Matter
Upper Complexity Bound Lim-ref Infinitely Finitely divisible Lim-ver . . . fin . . . inf fin I say inf when you let me cut.
Lower Complexity Bound Lim-ref -Lim-ver Infinitely Finitely divisible I let you cut when you say fin. Lim-ver -Lim-ref . . . inf fin . . . inf
Purely Formal Analogue Lim-ref -Lim-ver Infinite domain Finite domain 666 Lim-ver -Lim-ref 23455 21456
Purely Formal Analogue Lim-ref -Lim-ver Infinite domain Finite domain Lim-ver -Lim-ref I halt on another input each time he says fin in my opry oary simulation of him ganderin’ at my program. 666 23455 21456
Analogies Finite divisibility Finite domain . . . Effective empirical . . Ideal empirical Effective formal
Mixed Example: Penrose Computable Uncomputable ? ? ? . Human subject Cognitive scientist
Empirical Irony Computable Uncomputable You can verify human computability in the limit… UCUCCCC … Human subject Cognitive scientist
Empirical Irony Computable Uncomputable But only if you aren’t computable! UCUCUCCUCCU… Human subject Cognitive scientist
Computability Ideal Lim-ver Lim-ref Lim-dec
Computability We can verify our computability in the limit… only if we are not computable! Computable Lim-ver Lim-ref Lim-dec
Gold/Putnam Even assuming computability, you can converge to the true computable behavior only if you are not computable! Total computable functions f 0 f 32 f 243 . . .
Uncomputable Predictions (with Oliver Schulte) T -T • There exists T such that • T makes a unique prediction at each stage; • The predictions are very uncomputable…
Uncomputable Predictions (with Oliver Schulte) T -T • There exists T such that • T makes a unique prediction at each stage; • The predictions are very uncomputable… • But T is refutable by a computable method!
Moral 1: Rational Hobgoblins Ideal rationality No consistent computable convergent Exists inconsistent computable refuting Exists consistent method. computable non-convergent method. So “a foolish consistency” precludes truth-
Moral 2: Coup de Grace • The very idea of insulating deduction from empirical data restricts the truth-finding power of effective science. Empirical data Theorem Prover Yes n Q ?
Moral 2: Coup de Grace • The very idea of insulating deduction from empirical data restricts the truth-finding power of effective science. Empirical data Theorem Prover Yes n Q n
Q Conclusions • Justification = truth-finding performance. • Performance depends on problem complexity. • Formal and empirical complexity are similar. • Computational epistemology is unified • Standard epistemology is divided. • Insistence on division weakens truth-finding performance of effective science.
Closing Image Q Standard epistemology Q Computational epistemology
I’m rational Closing Image Q Standard epistemology Q Computational epistemology
I’m rational Closing Image Q Standard epistemology Q Computational epistemology
THE END
Pure Form of Naturalism actual world actual agent actual problem pure structure How does inquiry proceed? Have we found a correct answer? Will we? Are we guaranteed to? How might we improve? Is guaranteed success achievable in a completely specified, hypothetical problem? Which possible methods achieve solutions? Which maxims preclude solutions? What is necessary for guaranteed success to be achievable?
Every Problem in Its Place more “revolutionary”. Puzzle-solving adequacy grad ref Infinity (Kant’s Antinomy) Quantized conservation Uniformitarian geology Parallel postulate Probability, given existence lim ref Phenomenal laws cert ref grad ver Probability existence Conservation laws lim ver lim dec Finiteness Computability Global warming Chaos (M. Harrell) Duhem’s problem cert ver Experimental effects cert dec more“normal”.
19 th c. Geology Advancement Uniformitarianism (steady-state world) Stonesfield mammals 1814 Advancement Catastrophism (progressively cooling world): Stonesfield mammals 1814 creation
19 th c. Geology Uniformitarianism lim ref schedule new violation. U C Early mammals schedule lim ver Catastrophism schedule future fossils, the schedule stands Early reptiles new schedule
6. The Reliability Regress Reliability = success over a broad range of possibilities. But but how can we tell we are in the range of reliability? “method succeeds” ? actual input stream
Answer: Check with Another Method! H Same inputs to all 1 1 succeeds 2 2 succeeds 3 3 succeeds
Popperian Regress Solved H Methods never 1 take back a 1 succeeds rejection. 2 2 succeeds 3 3 succeeds … Later methods are as reliable as earlier. H Refutes H as reliably as any method in the regress
Verification Regress Solved H Later methods are as reliable as earlier. 1 1 succeeds 2 2 succeeds 3 3 succeeds … Methods never take back an acceptance. H Refutes H in the limit as reliably as any method in the regress.
Frequentist Probability n Background: a probability exists. n Hypothesis: its value is exactly p. p =p Observed freq too close to p “too close” set closer lim ref Observed freq too close to p “too close” set closer
Duhem’s Problem n Ref Background: there exist correct auxiliaries with respect to which isolated H is refutable. H H Refuted with auxiliaries New auxiliaries Ver lim ver Refuted with auxiliaries New auxiliaries
Antinomy of Pure Reason …neither [the world’s] finitude nor its infinitude can be contained in experience, because [the following are impossible]: 1. experience … of an infinite space…, 2. or again, of the boundary of the world by an empty space … Prolegomena, Section 52 c. ? Experience ?
Antinomy of Pure Reason Infinity lim ref position Intuition of more space. Infinite Finite lim ver Boundedness position intuitions, the position is not extended. more wait for more
Parallel Postulate ? lim ref pp intersect new pair
Paradigm Choice Ref Puzzle-solving adequacy: some articulation resolves by its standards all puzzles that arise within it. ad ad “unsolvable puzzle” Rearticulate Ver lim ver “unsolvable puzzle” Rearticulate
Paradigms “Internal Puzzle-solving adequacy”: articulation such that puzzle that arises within the paradigm Stage by which diligent effort would resolve the puzzle by the paradigm’s standards without rearticulation. ad A 1 solves 1 st puzzle A 1 solves 2 nd puzzle A 1 never solves 3 rd puzzle grad ref A 1 A 2 A 3
Kuhnian Moral gap lim ref lim ver Gradual refutability of each alternative does not suffice for convergence to an effective paradigm!
Conservation “Q is conserved in each reaction”: reaction stage at which the Q in = the Q out and Later stage, no more Q is observed in or out C Grad ver R 1 unbalanced R 1 R 2 unbalanced R 2 R 3 C
Cognitivism Ref Ver lim ver C C mismatch new program Computability program such that input the program produces the right output. mismatch new program
“Empirical Irony of Cog Sci” Computability program such that input stage by which the right output is produced. Prog 1 outputs 1 st datum C computer grad ref ideal Prog 2 outputs 2 nd datum Prog 3 never outputs 3 rd datum Prog 1 Prog 2 Prog 3 C
Transcendental Deductions • What kind of problem structure is necessary for a given sort of success?
Certain Verifiability and Refutability Certain verifiability: H is a union of “cones” of serious possibilities. Certain refutability: H is an intersection of cones of serious possibilities. “blue will be seen. ” cone = all possible extensions. . . “green forever”
Sextus Empiricus! Non-verifiability: Some input stream in H never enters a fan included in H. H H H. . . H
Limiting Verifiability Limiting verifiability: H is a countable union of intersections of fans. “The inputs will eventually stabilize to blue. ”
Transcendental System grad ver grad ref lim ver lim ref lim dec cert ref cert ver cert dec
Computable Case Similar! grad ver grad ref lim ref lim ver lim dec cert ref cert ver cert dec Turing-decidable relations
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